χ (3) non-Gaussian state generation for light using a trapped ion
aa r X i v : . [ qu a n t - ph ] O c t χ (3) non-Gaussian state generation for light using a trapped ion Magdalena Stobi´nska ∗ Institute f¨ur Optik, Information und Photonik, Max-Planck Forschungsgruppe, Universit¨at Erlangen-N¨urnberg,G¨unter-Scharowsky-Str. 1, Bau 24, 91058, Germany
G. J. Milburn
Centre for Quantum Computer Technology and School of Physical Sciences,The University of Queensland, St Lucia, Queensland 4072, Australia
Krzysztof W´odkiewicz † Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, Warszawa 00–681, Poland andDepartment of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131-1156, USA (Dated: December 20, 2018)According to the Gottesmann-Knill theorem the non-Gaussian states are necessary component fora nontrivial quantum computation. We show two efficient and deterministic methods of χ (3) non-Gaussian state generation for a cavity mode using a single trapped ion. Both require ion motionalstate transfer to the cavity field. The methods are experimentally feasible. The first is based onthe well-known protocol for an ion finite motional superposition state generation. It allows for anarbitrary good approximation of χ (3) non-Gaussian states. We give criteria based on the Wignerfunction which quantify the error resulting from the approximation. The second and novel methodenables an exact non-Gaussian state generation using one laser pulse only. I. INTRODUCTION
The Gottesmann-Knill theorem for continuous vari-ables (CV) states that quantum computing based onlyon components described by one- or two-mode quadraticHamiltonians, Gaussian states input, and measurementson canonical variables can be efficiently simulated by aclassical computation. That means, although some of thealgorithms are of fundamentally quantum nature, theydo not provide any speedup over classical processes [1].It has also been proved that construction of a CV uni-versal quantum computer for transformations that arepolynomial in those variables requires cubic or highernon-linear operations [2]. Therefore, investigation of thenon-Gaussian transformations and states generation iscrucial for a nontrivial quantum computation. More re-cently it has been noted that Kerr-like nonlinearities, in avariety of systems, enable high precision quantum metrol-ogy that would otherwise require entanglement to achieveit [3].Over the last decade ion trap experiments have ledthe emerging technologies of coherent quantum control,especially in quantum information theory [4] and quan-tum computation [5]. Those applications require efficientcreation and precise manipulation of both electronic andmotional trapped ion state. Various theoretical propos-als on how to produce nonclassical arbitrary states of ionmotion have been discussed. In experiment, Fock numberstates [6], coherent states [7], vacuum squeezed states [8],and Schr¨odinger cat states [9] have been realized. In this ∗ Electronic address: [email protected] † Electronic address: [email protected] latter case, the state is an entangled state of the vibra-tional and electronic degrees of freedom. On the contrary,in this paper we give a deterministic way to prepare thevibrational degree of freedom in a non-Gaussian statethat is not entangled with the electronic states. Accord-ing to our knowledge, neither non-Gaussian state (otherthan an entangled cat state) nor a superposition of morethan two coherent states has been observed so far.In the case of photons there is no practical method ofnon-Gaussian state generation so far. Efforts have beenmade to explore a class of the χ (3) non-Gaussian statesproduced using a photon coherent state | α i interactingwith χ (3) Kerr nonlinearity in an optical fiber | Ψ( α, τ ) i = e − | α | ∞ X n =0 α n √ n ! e i τ n ( n − | n i . (1)This class of non-Gaussian states, parametrized by theunitless evolution parameter τ [10], is the most popu-lar one. The state (1) is known also as the Kerr state.In general, this is a highly nonclassical state and aftera certain time of evolution τ in the fiber its Wignerfunction would take negative values in the phase space[11]. However, the nonlinearity in a fiber, or any otherexperimentally achievable Kerr medium, is too small, χ (3) ≃ −
22 m V , to reach a highly nonlinear regime andthus produce the negativity in an experimentally reason-able time, before it is destroyed by dissipation [12]. Al-though microstructured fibers seem to be more promisingwith χ (3) ≃ −
16 m V , their length does not exceed 1 mwith current technology.The most known examples of the Kerr state are the catstates e − iπ/ | iα i + e iπ/ | − iα i corresponding to τ = π ,which have been found useful for studies of quantum de-coherence and quantum-classical boundary [13, 14]. Thelarger cat states for which the two components | iα i and | − iα i are almost orthogonal ( α > .
5) find their appli-cation in quantum information processing [15] and quan-tum computation [16, 17].Recently, there has been introduced a probabilisticmethod of non-Gaussian state generation relying on aconditional photon subtraction from a squeezed vacuumstate [18, 19, 20, 21]. Such a state is a good approxi-mation for the cat state if the amplitude is small α < α > . χ (3) non-Gaussian state generation forlight using a single trapped ion. Both methods requireion motional state transfer to the cavity mode. The firstmethod is based on the well-known protocol [27] for anion finite motional superposition state generation. It al-lows to produce the χ (3) non-Gaussian states with ar-bitrary good approximation. We give criteria, based onWigner function comparison and its measurement preci-sion, which quantify the error resulting from the approx-imation. The second method is novel and it enables anexact non-Gaussian state generation using one laser pulseonly. We point out that a Wigner function measurementof ion motional state can be performed using currentlyavailable technology and already existing experimentalschemas. We also suggest a quantum metrology applica-tion, based on the work of Caves and co workers [3].This paper is organized as follows. In section II weshow that, applying a well-known protocol, one can pro-duce an approximated χ (3) non-Gaussian state of motionfor a trapped ion. The method relies on using a seriesof laser pulses to couple electronic and vibrational de-grees of freedom to effect the desired state preparationfor the motional degree of freedom. We give the criteriafor quantifying the extent to which the prepared state ap-proximates the desired non-Gaussian state, and discussthe technical limitations of the method. In section III wepresent an alternative, and novel, method for generationof an exact χ (3) non-Gaussian state. We also discuss therange of application of this method. We finish the pa- per with conclusions and a brief discussion of possibleapplications to quantum metrology. II. SERIES OF LASER PULSES METHOD
An ion in a Paul trap [28] may be prepared in an ar-bitrary state of the form | Ψ i = δ | Ψ e i| e i + β | Ψ g i| g i , (2)using the method proposed in [27, 29], where | Ψ e i = M X n =0 w en | n i , | Ψ g i = M X n =0 w gn | n i (3)are finite superpositions of ion motional states ( M < ∞ ), | n i is a Fock state of a harmonic oscillator potentialin the trap, | g i and | e i are the ion electronic groundand excited states respectively. Parameters δ and β arecomplex numbers obeying the normalization constraint | δ | + | β | = 1 and are set at the start of the experiment.The method is based on applying a series of alternatinglaser pulses tuned, first to the carrier, and then the redsideband, transition of the trapped ion. It works bothwithin and beyond the Lamb-Dicke regime. The ion isinitially prepared in its electronic and vibronic (motional)ground state | , g i . Adjusting the Rabi frequencies andduration of each pulse properly, one could achieve w en = w gn = w n equal to w n = 1 qP Mk =0 | α | k k ! α n √ n ! e i τ n ( n − . (4)These coefficients correspond to the coefficients of aquantum non-Gaussian state resulting from the unitaryevolution of a self-Kerr interaction (1) decomposed in theFock basis, up to the normalization factor (which resultsfrom the fact that we cut off the infinite sum in (1) andtake into account only the first M terms).This method produces an approximated χ (3) non-Gaussian state | Ψ ( M ) ( α, τ ) i corresponding to the statereached via a Kerr interaction with an arbitrary value ofevolution parameter τ . Furthermore the effective valueof τ can be much larger than can be achieved via unitaryinteraction under a realistic optical Kerr interaction werethe bosonic degree of freedom an optical field mode. Aproper choice of the cut-off value M enables one to ap-proximate the state (1) arbitrarily closely. If we set δ = 0for simplicity, the ion state is given by | Ψ ( M ) ( α, τ ) i = 1 qP Mk =0 | α | k k ! M X n =0 α n √ n ! e i τ n ( n − | n i| g i . (5)The number of required pulses for preparing the state (5)is equal to 2 M | Ψ ( M ) ( α, τ ) i = R M C M − · ... · R C | , g i , (6) n w n
14 15 16 17 180.0000.0020.0040.0060.0080 10 20 30 40 50 60 n w n
49 50 51 52 530.0000.0010.0020.0030.004
FIG. 1: The coefficients w n of a χ (3) non-Gaussian state de-composition in a Fock basis evaluated for α = 2 – the topfigure and α = 5 – the bottom figure. The dots above thered dashed line take values greater or equal to 10 − . If α = 2( α = 5) the significant coefficients range from w ( w ) to w ( w ). where C j and R j denote a carrier and a red sidebandlaser pulse respectively. Therefore M should be as smallas possible [30].The hint for existence of a good non-Gaussian stateapproximation comes from a simple observation that onlya finite amount of coefficients w n contribute to the sum(1) significantly and the exact number of them is strongly α dependent. The coefficients, evaluated for exemplaryvalues of the amplitude, α = 2 and α = 5, are depictedon Fig. 1.For a given value of an amplitude α the choice of thecut-off value M is based on three criteria. All criteria relyon comparing the Wigner function W ( M ) ( τ = 2 π, γ, γ ∗ )of an approximated state (5) with the untruncated one W ( τ = 2 π, γ, γ ∗ ) evaluated for (1). Please note that thecomparison is made for τ = 2 π . We chose this particularvalue of the evolution parameter because for this valuethe original Wigner function is given by a simple analyticformula, which can be computed directly.The first criterion derives from an investigation of theapproximated Wigner function isolines. The smaller thevalue of M the more the function deviates from an ideal Gaussian function for τ = 2 π : the isolines are no longercircles. The isolines of interest can be chosen arbitrarily.We however chose 0 .
1, 0 .
3, 0 . α,
0) separately. For anideal circle the ratio is always equal to 1.In order to take into account the interference re-sulting from the approximate state, which takes thevalues around zero, we calculate the maximal andaverage ratio of the difference π | W ( τ = 2 π, γ, γ ∗ ) − W ( M ) ( τ = 2 π, γ, γ ∗ ) (cid:12)(cid:12) to the value of the ideal Wignerfunction, for all points γ in a phase space. Therefore, thesecond criterion gives the maximal and average relativeerror of the approximation respectively.The third criterion reveals the percentage of points γ in a phase space for which W ( M ) ( τ = 2 π, γ, γ ∗ ) = W ( τ =2 π, γ, γ ∗ ) for a given precision. We assume a precision oforder of either ± − or ± − to be good enough, sinceit relates to the accuracy of both the Wigner functioncomputer visualisation using the density plots and theWigner function reconstruction using quantum tomogra-phy [31].The above criteria turn out to be more subtle than thewell-known fidelity F which measures the overlap of theapproximated and the original state in the phase space F = (cid:12)(cid:12)(cid:12) h Ψ ( M ) ( α, τ ) | Ψ( α, τ ) i (cid:12)(cid:12)(cid:12) = e − | α | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X m =0 | α | m m ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7)Obviously, the fidelity approaches unity in the limit of M → ∞ .We would like to point out that having the ion alreadyprepared in state (5) one could measure the Wigner func-tion of its vibronic state using either the standard methodof quantum tomography [32, 33] or the method of directmeasurement developed in [34]. This would allow for de-tailed investigation of χ (3) non-Gaussian states, whichhas never been verified experimentally so far.The direct Wigner function measurement method is es-pecially interesting. It relies on the fact that the Wignerfunction of a displaced vibronic state of an ion is relatedto the probability of finding the ion in the ground and theexcited state. The probabilities are measured by detect-ing a fluorescence signal. Such a measurement has beendone for a light in a cavity with accuracy of ± . Example: | Ψ ( M ) ( α = 2 , τ ) i generation Estimation of the minimal number of laser pulses for | Ψ ( M ) ( α = 2 , τ ) i generation, which approximates theoriginal state | Ψ( α = 2 , τ ) i for any value of the evolutionparameter τ well, requires comparison of the approxi-mated Wigner functions W ( M ) ( τ = 2 π, γ, γ ∗ ) evaluatedfor a few different cut-off values M , with the originalWigner function W ( τ = 2 π, γ, γ ∗ ). The selection of thebest possible cut-off values is based on analysis of sig-nificant coefficients w n . For α = 2 only the coefficientsranging from w to w are greater than 10 − . The co-efficients w to w are of order of 10 − . Therefore, wetest 10 ≤ M ≤ ,
0) for the isolines (0 .
1, 0 .
3, 0 .
5) ofthe approximated Wigner functions W ( M ) ( τ = 2 π, γ, γ ∗ ).The minimal value of M for which the ratios evaluatedfor W ( M ) ( τ = 2 π, γ, γ ∗ ) are equal to the ratios evaluatedfor the original Gauss function using the grid (discretizedphase space) with a step equal to ∆ γ = 0 .
04 is equal to14. It means that for the given grid and greater valuesof M , the numerical simulations of the Wigner functionwill not differ. Therefore, the value of M = 14 could beregarded as an appropriate cut-off value.The series of 28 laser pulses is experimentally feasible[30]. However, decreasing the quality of approximationonly a little, one can diminish the number of pulses to 20.The ratios evaluated for M = 10 ( M = 9) and isolines0 .
1, 0 . . M = 9 the error is equal to1 .
67 %. The maximal error is equal to 6 .
39 % (11 .
30 %).Within a given precision 10 − for M = 10 ( M = 9) thereare around 76 % (64 %) points for which the numericallyobtained Wigner function values are equal to the values ofthe ideal function. For the precision of 10 − and M = 10( M = 9) there are 36 % (26 %) of such points.These results show that M = 10 is still a good ap-proximation for the Wigner function analysis and mea-surement. The approximated functions for M = 10 and M = 14 with marked isolines are depicted on Fig. 2. Wealso include the plot for M = 9 for comparison.The fidelities evaluated for | Ψ (9) ( α = 2 , τ ) i , | Ψ (10) ( α =2 , τ ) i and | Ψ (14) ( α = 2 , τ ) i are equal to F = 0 . F = 0 . F = 0 . | Ψ (10) ( α = 2 , τ = π/ i , which is a superposition of fourcoherent states ϕ Rn t Rn R π .
99 ms R .
47 0 .
39 ms R .
23 0 .
35 ms R .
26 0 .
44 ms R .
00 0 .
47 ms R .
82 0 .
55 ms R .
21 0 .
55 ms R − .
30 0 .
75 ms R − .
95 0 .
81 ms R .
23 1 .
37 ms ϕ Cn t Cn C .
00 2 . µ s C − .
83 1 . µ s C .
33 1 . µ s C .
41 2 . µ s C − .
05 1 . µ s C − .
86 2 . µ s C − .
97 1 . µ s C − .
93 2 . µ s C − .
09 2 . µ s C − .
19 1 . µ sThese quantities were evaluated for the following Rabifrequencies Ω C = 1 MHz, Ω R = 100 kHz corresponding FIG. 2: The Wigner function for an approximated χ (3) non-Gaussian state: | Ψ (9) ( α = 2 , τ = 2 π ) i – the top figure, | Ψ (10) ( α = 2 , τ = 2 π ) i – the top middle figure, | Ψ (14) ( α =2 , τ = 2 π ) i – the bottom middle figure, | Ψ( α = 2 , τ = 2 π ) i –the bottom figure. FIG. 3: The Wigner function for an approximated χ (3) non-Gaussian state: | Ψ (9) ( α = 2 , τ = π ) i – the top figure, | Ψ (10) ( α = 2 , τ = π ) i – the middle figure, | Ψ( α = 2 , τ = 2 π ) i – the bottom figure. to carrier transition and red sideband respectively, andLamb-Dicke parameter η = 0 .
02. The total time of statebuild-up is equal to t c ≃ . M = 10 is is presented on Fig. 3. We alsopresent the Wigner functions for M = 9 and the originalone for | Ψ( α = 2 , τ = π ) i for comparison. Technical limitations
At the end of this section we estimate the maximumvalue of the amplitude α for which the method works.The maximum value seems to be α ≃ . . M corre-sponds to the maximal ion excitation | M i . The | M = 17 i is the upper limit for trapping the ion so far [36]. III. ONE LASER PULSE METHOD
The one laser pulse method of χ (3) non-Gaussian stategeneration requires one carrier resonance pulse appliedto the ion cooled down to a Lamb-Dicke regime. The vi-bronic state of the ion is initially prepared in a coherentstate | α i . The advantage of this method over the first oneis that it allows for generation of the original Kerr state(1) without any approximations. However, not all val-ues of the evolution parameter are accessible: the pulseduration is limited due to the ion decoherence.The interaction between an ion and a laser pulse ofRabi frequency Ω is governed by the following Hamilto-nian H = ¯ h Ω2 n σ + e iη ( ae − iν + a † e iν ) + h.c. o , (8)where σ + is an electronic state rising operator, a is avibronic state annihilation operator, ν is a trapping fre-quency, and η is a Lamb-Dicke parameter. Using theexpansion of the exponens function into a Taylor series e iη ( ae − iν + a † e iν ) = L X k =0 ( iη ) k k ! ( ae − iν + a † e iν ) k , (9)in the rotating wave approximation and in an interactionpicture the Hamiltonian is approximated by the first fiveterms of the expansion H rwaint = ¯ h Ω2 (cid:26) − η η (cid:18) − η + η (cid:19) a † a + η a † a (cid:27) ( σ + + σ − ) . (10)The terms in the first line in the above formula governthe free ion evolution. The last term in (10) we asso-ciate with a self-Kerr interaction Hamiltonian, up to theelectronic state operators. Therefore, the unitary evo-lution operator resulting from the nonlinear part of theHamiltonian is given by U ( t ) = e i Ω η t a † a ( σ + + σ − ) . (11)It allows for reading out the effective value of the evolu-tion parameter τ eff = Ω η t . (12)In the further discussion we neglect the electronic stateevolution since it may be made to take no part in thedynamics. To do this one first needs to prepare the elec-tronic state in an eigenstate of σ x using a π/ α . The laser pulseduration required for the Kerr state generation is thesame for all values of the amplitude. In other words,the time required for generation e.g. | Ψ( α = 2 , τ ) i and | Ψ( α = 5 , τ ) i is the same. However, accessibility of theevolution parameter τ eff is limited instead. The greaterRabi frequency Ω and Lamb-Dicke parameter value arethe shorter pulse duration t must be to obtain τ eff . Onthe other hand, the ion has to remain within the Lamb-Dicke regime in order to hold the expansion (10) true.Figure 4 shows the time of the pulse duration assum-ing its Rabi frequency Ω = 10 MHz and the Lamb-Dickeparameters η = 0 . η = 0 . η = 0 .
02, and η = 0 . µ s, which is within the coherence time of vibronicion state (190 ms), to obtain τ eff = π and therefore pro-duce the cat state. For the two remaining η values theduration of the pulse is of order of tenth of second.The duration of the pulse required for | Ψ( α, τ = 0 . i generation is equal to t = 0 .
16 ms for η = 0 . t =1 . µ s for η = 0 . t = 0 . η = 0 .
02 and t = 0 . µ sfor η = 0 . τ = 0 .
04 the first negativities in theWigner function become visible [11].This method allows for obtaining also the other coher-ent state superpositions easily. For example, for τ eff = π one achieves superposition of six states and if τ eff = π oneachieves superposition of four states (the compass state)[37]. For η = 0 . | Ψ( α, τ = π/ i the pulse durationis equal to t = 51 . µ s, and for | Ψ( α, τ = π/ i it isequal to t = 77 . µ s. These values are experimentallyfeasible.The one laser pulse method of χ (3) non-Gaussian stategeneration is limited due to introduced cut-off in the ex-pansion (9). It is valid only if we take the expansionfor a small parameter. To estimate the small parameterwe approximate the quantum operators by the classicalamplitudes a ≃ αe iη ( ae − iν + a † e iν ) = L X k =0 (2 iη | α | ) k k ! (cid:18) ae − iν + a † e iν | α | (cid:19) k (13)and read out the small parameter in the expansion to be2 ηα . The above series is decreasing if ηα < . This constraint is enough to ensure that the contribu-tion from the higher order terms is negligible. For exam-ple, for α = 5 and η = 0 .
1, obeying α = η ( η = 0 .
09 and α < η ) the coefficients (2 η | α | ) k k ! are equal to: 0 . . k = 2, 0 .
041 (0 . k = 4, 0 . . k = 6. For coefficients obeying α < η the sixth ordercoefficient is of two orders of magnitude smaller than theforth order coefficient. IV. WEAK FORCE DETECTION
We now show how the cat-state weak force detectionprotocol proposed in Munro et al [38] can be implementedin an ion trap given the ability to engineer an approxi-mate cat state as we have described. It must be admittedthat there is no compelling case to use the vibrational mo-tion of a trapped ion as a weak force detector. However,the method we propose here would be a nice demonstra-tion of how non-Gaussian states can beat the standardquantum limit for weak force detection.Suppose one was able to prepare the vibrational degreeof freedom in the non-Gaussian state | ψ i i = 1 √ (cid:16) e iπ/ | α i + e − iπ/ | − α i (cid:17) (14)with the amplitude α real. When a weak force acts it canbe described by the action of the unitary displacementoperator D ( iǫ ) = exp( iǫa † − iǫa ) acting on the initial state | ψ i i to give the output state | ψ o i = D ( iǫ ) | ψ i i . Using theresult that D ( iǫ ) | α i = e i Im( iαǫ ) | α + iǫ i (15)we find that for ǫ ≪ α , and α real that | ψ o i = cos( π/ αǫ ) | + i + i sin( π/ αǫ ) |−i (16)where the even and odd parity states are given by |±i = 1 √ | α i ± | − α i ) (17)In other words, the weak force is well approximated bya rotation in the two dimensional parity subspace. As in[38] it then follows that the minimum detectable force isthen given by ǫ min = 12 α (18)which beats the standard quantum limit by a factor of( α ) − .Ion trap provide a simple way to reach this lowerbound. Suppose that after the weak force has acted, thetotal vibrational and electronic state is given by Eq. (16).The first step is to make a π/ R = e − iπa † aσ z (19)which can easily be done [39]. Finally another π/ | ψ o i = cos( π/ αǫ ) | + i| g i + i sin( π/ αǫ ) |−i| e i . (20)The electronic state can now be readout, and the proba-bility to find the ion in, say, the excited state is P + = 12 (1 − sin(2 αǫ )) . (21)Sampling this distribution gives the estimation for theforce with a minimum detectable force that is inverselyproportional to α . For example, the minimum force re-quired to shift the interference distribution by one fringeis ǫ min = π/ (4 α ). V. CONCLUSIONS
In the paper we have presented two experimentally fea-sible methods of χ (3) non-Gaussian states generation forlight using a trapped ion. The first method is based on awell-known protocol (a series of laser pulses) and allowsfor generation of an arbitrarily well approximated non-Gaussian state. The approximation is quantified by three criteria based on Wigner function analysis. This methodis only limited by technical parameters of the trap: thebinding energy.The second method enables an exact non-Gaussianstate generation using one laser pulse. It is limited by thedecoherence time of the motional ion state. However, ad-justing the laser pulse and trap parameters properly oneis able to produce a cat state.Based on the proposed protocols [23, 24, 25] we believethat non-Gaussian state transfer from ion motion to alight beam is possible in a foreseeable future. It willenable application of a nontrivial quantum computingprotocols based for example on coherent states. This isalso a step towards exploring so far unknown branch ofquantum optics such as non-Gaussian states. Acknowledgments
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Isoline M 9 10 11 12 13 14 15 160.1 1.33 (33 %) 1.20 (20 %) 1.10 (10 %) 1.04 (–) 1.01 (–) 1.01 (–) 1.00 (–) 1.00 (–)0.3 1.23 (23 %) 1.10 (10 %) 1.04 (–) 1.01 (–) 1.00 (–) 1.00 (–) 1.01 (–) 1.01 (–)0.5 1.19 (19 %) 1.08 (8 %) 1.01 (–) 1.04 (–) 1.04 (–) 1.04 (–) 1.04 (–) 1.04 (–)TABLE I: The ratios of the most to the least distant point, with respect to the point (2 , .
1, 0 .
3, 0 .
5, ofthe approximated Wigner function W ( M ) ( τ = 2 π, γ, γ ∗ ) for 9 ≤ M ≤ PPPPPPP
Accuracy M 9 10 11 12 13 14 15 1610 −
64 % 76 % 90 % 99 % 100 % 100 % 100 % 100 %10 −
26 % 36 % 44 % 53 % 65 % 80 % 97,% 100 %TABLE II: The number of points γ in the phase space for which W ( M ) ( τ = 2 π, γ, γ ∗ ) = W ( τ = 2 π, γ, γ ∗ ) at a given accuracyfor 9 ≤ M ≤ ❍❍❍❍❍ Error M 9 10 11 12 13 14 15 16Average 1.67 % 0.98 % 0.55 % 0.30 % 0.15 % 0.08 % 0.04 % 0.02 %Maximal 11.30 % 6.39 % 3.48 % 1.84 % 0.94 % 0.47 % 0.22 % 0.11 %TABLE III: The average and the maximal error estimated for α = 2 and for 9 ≤ M ≤ Τ @ ms D Τ @ Μ s D Τ @ s D Τ @ s D FIG. 4: The time of pulse duration required for Kerr state | Ψ( α, τ ) i generation for Lamb-Dicke parameter η = 0 . η = 0 . η = 0 .
02 – thebottom middle figure, η = 0 ..